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doeboy

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[求助] 求高人指点一个常微分方程问题

如图

这个方程的解是什么?是数学物理里面的某个特殊函数吗?万分感谢!!
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dbb627

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【答案】应助回帖


感谢参与,应助指数 +1
doeboy: 金币+1, 有帮助 2012-04-18 10:37:28
y是x的函数,还是t的函数
>> y=dsolve('D2y=x^2*y','x')

y =

(C6*besselk(1/4, x^2/2)*(x^2)^(1/2))/(pi^(1/2)*x^(1/2)) + (pi*2^(1/2)*4^(1/4)*C5*besseli(1/4, x^2/2)*(x^2)^(1/2))/(4*x^(1/2)*gamma(3/4))

>> y=dsolve('D2y=x^2*y','t')

y =

C2/exp(t*x) + C3*exp(t*x)
The more you learn, the more you know, the more you know, and the more you forget. The more you forget, the less you know. So why bother to learn.
2楼2012-04-18 10:32:36
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doeboy

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引用回帖:
2楼: Originally posted by dbb627 at 2012-04-18 10:32:36:
y是x的函数,还是t的函数
>> y=dsolve('D2y=x^2*y','x')

y =

(C6*besselk(1/4, x^2/2)*(x^2)^(1/2))/(pi^(1/2)*x^(1/2)) + (pi*2^(1/2)*4^(1/4)*C5*besseli(1/4, x^2/2)*(x^2)^(1/2))/(4*x^(1/2)* ...

y是x的函数
3楼2012-04-18 10:36:54
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dbb627

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【答案】应助回帖

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doeboy: 金币+9, ★★★★★最佳答案 2012-04-18 10:44:02
引用回帖:
3楼: Originally posted by doeboy at 2012-04-18 10:36:54:
y是x的函数

数学物理里应该解出贝塞尔函数
matlab解的
y=dsolve('D2y=x^2*y','x')

y =

(C6*besselk(1/4, x^2/2)*(x^2)^(1/2))/(pi^(1/2)*x^(1/2)) + (pi*2^(1/2)*4^(1/4)*C5*besseli(1/4, x^2/2)*(x^2)^(1/2))/(4*x^(1/2)*gamma(3/4))
The more you learn, the more you know, the more you know, and the more you forget. The more you forget, the less you know. So why bother to learn.
4楼2012-04-18 10:38:35
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doeboy

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引用回帖:
4楼: Originally posted by dbb627 at 2012-04-18 10:38:35:
数学物理里应该解出贝塞尔函数
matlab解的
y=dsolve('D2y=x^2*y','x')

y =

(C6*besselk(1/4, x^2/2)*(x^2)^(1/2))/(pi^(1/2)*x^(1/2)) + (pi*2^(1/2)*4^(1/4)*C5*besseli(1/4, x^2/2)*(x^2)^(1/2))/(4*x ...

恩,已经找到了,是Parabolic cylinder function,参见http://en.wikipedia.org/wiki/Parabolic_cylinder_function,多谢帮助!
5楼2012-04-18 10:43:56
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